Patterns in Coin Tosses

We discussed the following problem in class today:

Consider an infinite sequence of independent coin tosses of a fair coin and define the following two random variables:

Z1 = Number of coin tosses till the pattern HTT first appears.

Z2 = Number of coin tosses till the pattern HTH first appears.

Then, somewhat counterintuitively, E[Z1] and E[Z2] are not the same! Its a common error to assume that since the probability of 3 tosses appearing in either pattern is 1/8, the expected time of first occurrence is also the same. But the time of first occurrence also depends on wether different occurrences of the string can overlap. Here’s the talk by by Peter Donnelly that I mentioned – it discusses this problem and a bunch of other interesting ones.

As an exercise, you can try using the idea of conditional expectations to compute E[Z1] and E[Z2].

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